list rational zeros?

Question

Given f(x)=x^3+2x^2-13x+10, (a) list all possible rational zeros, (b) find all rational zeros, and (c) factor f(x).

Answer 1

a) Rational zeros = ±factors of 10 = ±1, ±2, ±5, ±10

 

b) f(-5) = f(1) = f(2) = 0

So, rational zeros are x = -5, 1, 2

 

c) f(x) = (x+5)(x-1)(x-2)

Answer 2

using synthetic division, possible zeroes are factors of 10 divided by factors of 1, the coefficient

of x^3, so we have +1, +2, +5, and +10

 

let's try 1

 

 

     1             1    2    -13    10

                          1       3   -10

                    1    3    -10     0   

                     

now divide x^3+2x^2-13x+10 by x-1                   

 

              

                   

 

                                x^2 + 3x - 10

                               ________________

                   (x-1)   / x^3+2x^2-13x+10

                            /   x^3-x^2

                              _________

                                       3x^2-13x

                                       3x^2-3x

                                      _________

                                              -10x+10

                                              -10x+10

                                              ________

                                                         0

  now factor x^2+3x-10 into (x+5)(x-2)

so we have (x-1)(x+5)(x-2) for the factorization and 1, -5, and 2 for the zeroes